Question

Let A, B, C be three points whose position vectors respectively are 

`vec"a" = hat"i" + 4hat"j" + 3hat"k"`

`vec"b" = 2hat"i" + αhat"j" + 4hat"k", α ∈ "R"`

`vec"c" = 3hat"i" - 2hat"j" + 5hat"k"`

If α is the smallest positive integer for which `vec"a", vec"b", vec"c"` are noncollinear, then the length of the median, in ΔABC, through A is ______.

विकल्प

• `sqrt(82)/2`

• `sqrt(62)/2`

• `sqrt(69)/2`

• `sqrt(66)/2`

Answer 1

Let A, B, C be three points whose position vectors respectively are 

`vec"a" = hat"i" + 4hat"j" + 3hat"k"`

`vec"b" = 2hat"i" + αhat"j" + 4hat"k", α ∈ "R"`

`vec"c" = 3hat"i" - 2hat"j" + 5hat"k"`

If α is the smallest positive integer for which `vec"a", vec"b", vec"c"` are noncollinear, then the length of the median, in ΔABC, through A is `underlinebb(sqrt(82)/2)`.

Explanation:

Given: position vectors of three-point A, B, C are

`vec"a" = hat"i" + 4hat"j" + 3hat"k"`

`vec"b" = 2hat"i" + αhat"j" + 4hat"k"`

`vec"c" = 3hat"i" - 2hat"j" + 5hat"k"`

Now `vec("AB") - vec"b" - vec"a" = hat"i" + (α - 4)hat"j" + hat"k"`

And `vec("AC") = vec"c" - vec"a" = 2hat"i" - 6hat"j" + 2hat"k"`

If A, B, C are collinear, then `vec("AB") || vec("AC")`

⇒ `1/2 = (α - 4)/(-6) = 1/2`

⇒ α = 1

∴ α = 2 is the smallest positive integer for which A, B, C are non-collinear.

Now, mid-point of BC = `"p"(5/2, 0, 9/2)`

∴ Length of the median through A = AP

= `sqrt((5/2 - 1)^2 + (4)^2 + (9/2 - 3)^2)`

= `sqrt(9/4 + 16 + 9/4)`

= `sqrt(82)/2`